# What Whole Number Areas Are Possible With Squares on Dot Paper?

I am getting better at picking good presentations based on their titles and descriptions. And, at the NCTM 2015 Boston Conference, I had a pretty high batting average. I have already written about Susan Jo Russell's talk and about Math Fun Facts shared by Francis Edward Su -- talks I really enjoyed. I also attended a presentation on NCTM's theme of "From Principles to Actions" where the speakers spent a lot of time talking about students' mathematical identities.

As part of the presentation they had us working to solve a problem on dot paper. I really liked the problem, but I thought it could be a bit more open and general. Here is my version:

#### Using a 5 x 5 grid of dot paper, what integer areas can you form using single squares?

Examples are provided below for clarity. Note that the vertices of the squares must be on the dots and that the square need not be orthogonal to the border. Here is blank dot paper if you would like it.

Ben

April 24, 2015@ 4:23 pmRather than post an entire solution, I thought I would make a related remark:

If you consider _rectangles_ that can be drawn in using the vertices and orthogonal to the border, then you will notice that the number corresponds to entries in a multiplication table.

For example, thinking just about the squares: In the 5x5 grid (which can be thought of as a 4x4 array) there are a total of 16 1x1 squares you could drawn in; this corresponds to the 16 in the bottom right hand corner of a 4x4 times table. Similarly, there is a total of 1 4x4 square (i.e., the whole thing!) you can drawn in; this corresponds to the 1 in the top left hand corner of a 4x4 times table. For mxn rectangles (m, n distinct) there will be (by symmetry) the same number of mxn and nxm rectangles; fittingly, you can count them out and place the respective totals symmetrically about the NW/SE diagonal.

It is a nice problem to figure out just _why_ this process works!

Tyler

April 24, 2015@ 6:16 pmBen,

That's another great problem I have solved in the past. You start with a 4x4 array and figure out how many total orthogonal squares can be made, then generalize to m*n. I have worked on that previously, but had not made the connection to the multiplication table as clearly as you did.